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\title{Mathematical Writing Exercise Chapter 02 (2.1-2.4)}
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%\date{2023 年 10 月 31 日}
%\date{March 9, 2021}

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\begin{enumerate}

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\item  %Problem 01
True or False. 
\begin{enumerate}[label={(\arabic*)}]
\item  What are the differences between theorems, lemmas, and propositions? 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%\item  
To some extent, the answer depends on the context in which a result appears.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Generally, a theorem is a major result that is of independent interest. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%\item  
The proof of a theorem is usually nontrivial. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A lemma is an auxiliary result -- a stepping stone towards a theorem. Its proof may be easy or difficult.
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
A straightforward and independent result that is worth encapsulating but that does not merit the title of a theorem may also be called a lemma. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

%\item  Indeed, there are some famous lemmas, such as the Riemann-Lebesgue Lemma in the theory of Fourier series and Farkas's Lemma in the theory of constrained optimization. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Whether a result should be stated formally as a lemma or simply mentioned in the text does not depend on the level at which you are writing. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  In a research paper in linear algebra it would be inappropriate to give a lemma stating that the eigenvalues of a symmetric positive definite matrix are positive, as this standard result is so well known; in a textbook for undergraduates, however, it would be sensible to formalize and prove this result. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 02
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  It is not advisable to label all your results theorems, because if you do so you miss the opportunity to emphasize the logical structure of your work and to direct attention to the most important results. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
If you are in doubt about whether to call a result a lemma or a theorem, call it a theorem. %lemma. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The term proposition is less widely used than lemma and theorem and its meaning is less clear. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
It tends to be used as a way to denote a minor theorem. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Lecturers and textbook authors might feel that the modest tone of its name makes a proposition appear less daunting to students than a theorem. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
However, a proposition is not, as one student thought, ``a theorem that might not be true.''
...\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A corollary is a direct or easy consequence of a lemma, theorem, or proposition. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  It is important to distinguish between a corollary, which does not imply the parent result from which it came, and an extension or generalization of a result. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Be careful not to over-glorify a corollary by failing to label it as such, for this gives it false prominence and obscures the role of the parent result. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 03
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  How many results are formally stated as lemmas, theorems, propositions, or corollaries is a matter of personal style. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Some authors develop their ideas in a sequence of formal statements and proofs interspersed with definitions and comments. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  At the other extreme, some authors state very few results formally. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
A good example of the latter style is the classic book The ``Algebraic Eigenvalue Problem'' by James Wilkinson, in which only four titled theorems are given in 662 pages. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  As Ralph Boas notes, ``A great deal can be accomplished with arguments that fall short of being formal proofs.'' 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 04
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  A fifth kind of statement used in mathematical writing is a conjecture -- a statement that the author thinks may be true but has been unable to prove or disprove. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
The author will usually have some strong evidence for the veracity of the statement. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A famous example of a conjecture is the Goldbach conjecture, which states that every even number greater than 2 is the sum of two primes; this is still unproved. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  One computer scientist (let us call him Alpha) joked in a talk ``This is the Alpha and Beta conjecture. If it turns out to be false I would like it to be known as Beta's conjecture.'' 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  However, it is not necessarily a bad thing to make a conjecture that is later disproved: identifying the question that the conjecture aims to answer can be an important contribution. 
...\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A hypothesis is a statement that is taken as a basis for further reasoning, usually in a proof -- for example, an induction hypothesis. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Hypotheses that stand on their own are uncommon; two examples are the Riemann hypothesis and the continuum hypothesis.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 05
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  Readers are often not very interested in the details of a proof but want to know the outline and the key ideas. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
They hope to learn a technique or principle that can be applied in other situations. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  When readers do want to study the proof in detail they naturally want to understand it with the minimum of effort. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
To help readers in both circumstances, it is important to emphasize the structure of a proof, the ease or difficulty of each step, and the key ideas that make it work. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Here are some examples of the sorts of phrases that can be used (most of these are culled from proofs by Beresford Parlett). 
\begin{itemize}
\item  The aim/idea is to ...
\item  Our first goal is to show that ...
\item  Now for the harder part.
\item  The trick of the proof is to find ...
\item  ... is the key relation.
\item  The only, but crucial, use of ... is that ...
\item  To obtain ... a little manipulation is needed.
\item  The essential observation is that ...
\end{itemize}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  When you omit part of a proof it is best to indicate the nature and length of the omission, via phrases such as the following. 
\begin{itemize}
\item  It is easy/simple/straightforward to show that ...
\item  Some tedious manipulation yields ...
\item  An easy/obvious induction gives ...
\item  After two applications of ... we find ...
\item  An argument similar to the one used in ... shows that ...
\end{itemize}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  You should also strive to keep the reader informed of where you are in the proof and what remains to be done. 
Useful phrases include
\begin{itemize}
\item  First, we establish that ...
\item  Our task is now to ...
\item  Our problem reduces to ...
\item  It remains to show that ...
\item  We are almost ready to invoke ...
\item  We are now in a position to ...
\item  Finally, we have to show that ...
\end{itemize}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  The end of a proof is often marked by the halmos symbol $\square$. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
Sometimes the abbreviation QED (for the Latin quod erat demonstrandum, meaning ``which was to be demonstrated'') is used instead.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

%There is much more to be said about writing (and devising) proofs.

%References include Franklin and Daoud [102], Gamier and Taylor [111], Lamport [213], Leron [215], and Polya [266].

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 06
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  A pedagogical tactic that is applicable to all forms of technical writing (from teaching to research) is to discuss specific examples after the general case. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  It is tempting, particularly for mathematicians, to adopt the opposite approach, but beginning with examples is often the most effective way to explain %(see Boas's article [42] and the quote from it at the beginning of this chapter, a quote that itself illustrates this principle!). 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  A good example of how to begin with a specific case is provided by Gilbert Strang in Chapter 1 of << Introduction to Applied Mathematics >>:

\begin{center}
\fbox{
\begin{minipage}{12cm}
The simplest model in applied mathematics is a system of linear equations. It is also by far the most important, and we begin this book with an extremely modest example:
\begin{eqnarray*}
2x_1 + 4x_2 &=& 2, \\ 
4x_1 + 11x_2 &=& 1.
\end{eqnarray*}
\end{minipage}
}
\end{center}

After some further introductory remarks, Strang goes on to study in detail both this $2\times 2$ system and a particular $4\times 4$ system. General $n\times n$ matrices only appear several pages later. 
...\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Another example is provided by David Watkins's << Fundamentals of Matrix Computations >>. 
Whereas most linear algebra textbooks introduce Gaussian elimination for general matrices before discussing Cholesky 
factorization for symmetric positive definite matrices, Watkins reverses the order, giving the more specific but algorithmically more straightforward method first.
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 07
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  Three questions to be considered when formulating a definition are why, where, and how.
First, ask yourself why you are making a definition: is it really necessary? 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Inappropriate definitions can complicate a presentation and too many can overwhelm a reader, so it is wise to imagine yourself being charged a large sum for each one. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Instead of defining a square matrix $A$ to be contractive with respect to a norm $\lVert\cdot\rVert$ if $\lVert A \rVert < 1$, which is not a standard definition, you could simply say ``$A$ with $\lVert A \rVert < 1$'' whenever necessary. 
This is easy to do if the property is needed on only a few occasions, and it saves the reader having to remember what ``contractive'' means. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  For notation that is standard in a given subject area, judgment is needed to decide whether the definition should be given. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  With symbols, potential confusion can not be avoided by including the corresponding words. 
For example, if $\rho(A)$ is not obviously the spectral radius of the matrix $A$ you can say ``the spectral radius $\rho(A)$''.
\dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 08
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  The second question is ``where?'' 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
The practice of giving a long sequence of definitions at the start of a work is highly recommended. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Ideally, a definition should be given in the place where the term being defined is first used. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
If it is given much earlier, the reader will have to refer back, with a possible loss of concentration (or worse, interest). 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Try to minimize the distance between a definition and its place of first use. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

%\item  It is not uncommon for an author to forget to define a new term on its first occurrence. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)

%\item  For example, Norman Steenrod uses the term ``grasshopper reader'' on page 6 of his essay on mathematical writing but he does not define it until it occurs again on the next page. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  To reinforce notation that has not been used for a few pages you may be able to use redundancy. 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
For example, ``The optimal steplength $\alpha^*$ can be found as follows.''
This implicit redefinition either reminds readers what $\alpha^*$ is or reassures them that they have remembered it correctly. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 09
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  Finally, how should a term be defined? 
%\dotfill (\,\,\,\,\,\,\,\,\,\,)
%
%\item  
There may be a unique definition or there may be several possibilities (a good example is the term $M$-matrix, which can be defined in at least fifty different ways). 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  You should aim for a definition that is long, expressed in terms of a fundamental property or idea, and consistent with related definitions. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  As an example, the standard definition of a normal matrix is a matrix $A \in \mathbb{C}^{n\times n}$ for which $A^*A = A A^*$ (where $*$ denotes the conjugate transpose). 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  There are at least seventy different ways of characterizing normality, but none has the simplicity and ease of use of the condition $A^*A = A A^*$. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 10
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  By convention, ``if'' means ``if and only if'' in definitions, so do not write ``The graph $G$ is connected if and only if there is a path from every node in $G$ to every other node in $G$. ''
Write ``The graph $G$ is connected if there is a path from every node in $G$ to every other node in $G$'' (and note that this definition can be rewritten to omit the symbol $G$). 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  It is common practice to italicize the word that is being defined: ``A graph is \textit{connected} if there is a path from every node to every other node.'' 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  This has the advantage of making it perfectly clear that a definition is being given, and not a result. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  This emphasis can also be imparted by writing ``A graph is defined to be connected if ...,'' ``A graph is said to be connected if ...,'' or ``A graph is called connected if ''. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\end{enumerate}

\vspace{0.2cm}

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\item  %Problem 11
True or False. 
\begin{enumerate}[label={(\arabic*)}]

\item  %If you have not done so before, 
It is instructive to study the definitions in a good dictionary. 
They display many of the attributes of a good mathematical definition: they are concise, precise, consistent with other definitions, and easy to understand. 
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Definitions of symbols are usually made with a simple equality, perhaps preceded by the word ``let'' if they are inline, as in ``let $q(x) = ax^2 + bx + c$''.  
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  Various other notations have been devised to give emphasis to a definition, including
\begin{eqnarray*}
q(x) &:=& ax^2 + bx + c, \\ 
ax2 + bx + c &:=& q(x), \\ 
q(x) &\overset{\text{def}}{=}& ax^2 + bx + c, \\ 
q(x) &\equiv& ax^2 + bx + c, \\ 
q(x) &\overset{\triangle}{=}& ax^2 + bx + c. 
\end{eqnarray*}
\dotfill (\,\,\,\,\,\,\,\,\,\,)

\item  If you use one of these special notations you must use it consistently, otherwise the reader may not know whether a straightforward equality is meant to be a definition.
\dotfill (\,\,\,\,\,\,\,\,\,\,)


\end{enumerate}

\vspace{0.2cm}

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%\item  %Problem 01
%True or False. 
%\begin{enumerate}
%\item  
%\item  
%\item  
%\item  
%\end{enumerate}
%
%\vspace{0.2cm}


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\end{enumerate}


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